Point Diffractors

We begin with a simple model consisting of nine point diffractors. There are two main reasons for choosing this model. First, point diffractors are a fundamental element of imaging, and can be used to build up any complex reflector. Second, they give a direct measure of imaging quality, in particular resolution.

In this model, nine point diffractors are embedded in a medium with a constant velocity of 2000 m/s. The model size is 500 m by 1000 m (Figure 3.4A). Three shots (Figure 3.4B, C and D) with a source signature corresponding to a 30 Hz Ricker wavelet located at 200 m, 500 m and 800 m on the surface and records are modelled for 101 receivers with 10 m trace spacing spread evenly on the surface of model. To generate the records, the finite-difference scheme expressed in Equation 2.11 with PML boundary condition is used.

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Figure 3.5 shows the results of applying MLSRTM and GDM to the first shot (Figure 3.4B) together with their associated amplitude spectra. Comparing Panels A, C and E, it can be seen that the result from MLSRTM is better than that for GDM. First, MLSRTM produces fewer artefacts: the GDM image (Panel A) contains quite strong arc-shaped artefacts which are not present in the MLSRTM result (Panels C and E). Second, the image of a single diffractor has fewer sidelobes. This is because the deconvolution imaging condition in MLSRTM compensates for the source wavelet whereas the zero-lag crosscorrelation imaging condition in GDM actually exaggerates the effect of the source signature. Finally, MLSRTM can produce higher spatial resolution images than GDM. This is supported by the amplitude spectra (the right column of Figure 3.5), which are calculated by applying the Fourier transform on the images along depth. From the panels, it can be seen that there is a larger bandwidth in Panel F than Panels D and B. This means the image in Panel E has higher vertical resolution than Panels A and C.

Figure 3.6 shows the corresponding images and spectra of the second shot (Figure 3.4C). From the results, MLSRTM increases the resolution and reduces the artefacts in the image. However, one interesting phenomenon is that only the images of the three point diffractors of the middle column are horizontal whilst the others are tilted. This is because the second shot illuminates the three point diffractors symmetrically. Furthermore, due to the tilting of the images, the diffractors directly under the source have higher resolution than the other diffractors.

Forward modelling of the images with Equation 3.14 produces the predicted records, which are presented in Figure 3.7. Since the first shot (Figure 3.4B) and the third shot (Figure 3.4D) are anti-symmetrical, only the first and second shots are displayed here. By comparing Panels A and B with Panels C to F in Figure 3.7, it is obvious that the GDM images do not match the true records shown in Figure 3.4 very well. First, the predicted data for GDM contain several events that do not exist in the true records, including events similar to the direct arrivals generated by the arc-shaped artefacts in Figure 3.5A. Second, the predicted data of the GDM images have more sidelobes than the true records. Nine hyperbolic events are clearly seen in the true records; however in the predicted data of the GDM images, the extra sidelobes of the events interfere with each other so that not all the hyperbolic events can be identified. Third, later arrivals in the predicted data of the GDM images

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are much weaker than the actual arrivals in the true records. This is because the GDM images of the deep diffractors are weaker when using a zero-lag crosscorrelation imaging condition, and cannot accurately recover the amplitude of reflectors. By contrast, the predicted data of the MLSRTM images are almost identical to the true records indicating that the MLSRTM images can “explain” the records. Furthermore, if the modelling algorithm precisely describes the propagation of actual waves in the records, then the migration images inverted using MLSRTM will be accurate. The shape of the images, however, is not identical to the actual model since the data have limited bandwidth. With the deconvolution imaging condition in MLSRTM, the spatial frequency spectra of the inverted images are significantly expanded (Panels B, D and F in Figures 3.5 and 3.6).

Figure 3.4 (A) Model containing nine point diffractors in a uniform medium with a constant velocity of 2000 m/s. (B), (C) and (D) Shot records for sources located at 200 m, 500 m and 800 m on the surface respectively. The source for each shot is a 30 Hz Ricker wavelet. Each shot has 101 traces with a trace spacing of 10 m.

For multi-shot migration, the GDM image is the stack of the individual images of each shot. However,

(A) _(B)

(D) (C)

77 the MLSRTM image is iteratively updated by using

ž@+1= ž@+ )â@,

where ž_@+1 is the updated image, ž_@ is the current image and â_@ is the image update and a stack of the contribution of each shot. The result of migrating the three shots are shown in Figure 3.8. As can be seen, the image obtained by migrating the three shots with GDM has fewer artefacts than the corresponding image with a single shot. This demonstrates that migrating more shots can suppress artefacts. However, comparing the GDM image of the three shots with the corresponding result from MLSRTM shows that MLSRTM has even fewer artefacts. The right column of Figure 3.8 shows the amplitude spectra of the images on the left. It is clear that the MLSRTM images have broader bandwidth than the GDM image. An interesting phenomenon is that the spectra of three column point diffractors have almost identical bandwidth, irrespective of their location, whereas the spectra of the single shot images depend on the location of each diffractor. Figure 3.9 shows the shot records corresponding implied in the GDM and MLSRTM results. As can be seen, the GDM images of three shots fit the data better than the GDM image of a single shot. However, the three shot GDM image still does not accurately fit the records.

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Figure 3.5 GDM (RTM) and MLSRTM images of the first shot (Figure 3.4B), the source of which is located at 200 m on the surface, and their amplitude spectra. (A) is the GDM image while (C) and (E) are MLSRTM images after 20 and 200 iterations respectively. (B), (D) and (F) are the amplitude spectra corresponding to the results shown on the left.

(C)

(A) (B)

(D)

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Figure 3.6 The same as Figure 3.5, but for the shot located at 500 m. (C)

(A) (B)

(D)

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Figure 3.7 The predicted shot records of the GDM (RTM) and MLSRTM images of the first and second shots (Figure 3.4B, C). (A) is the predicted first shot of the GDM image while (C) and (E) are the predicted first shot of MLSRTM images after 20 and 200 iterations respectively. (B), (D) and (F) are the corresponding predicted records for the second shot.

(C)

(A) (B)

(D)

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Figure 3.8 The GDM (RTM) and MLSRTM images of three shots. (A) is the GDM image while (C) and (E) are MLSRTM images after 20 and 200 iterations respectively. (B), (D) and (F) are the spectra corresponding to the images on the left.

(C)

(A) (B)

(D)

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Figure 3.9 The predicted shot records of GDM (RTM) and MLSRTM images of three shots. (A) and (B) are the predicted first and second shots of the GDM image respectively. (C) and (D) are the predicted first and second shots of the MLSRTM image after 20 iterations respectively while (E) and (F) are for 200 iterations.